Classification and general-metric solutions for higher-dimensional solvable algebras

Classify solvable Lie algebras of dimension six and higher and solve the algebraic constraint for a general symmetric bilinear form \(h\) on every classified solvable algebra, including the determination of all resulting signatures.

Background

The solvable sector is not classified in dimensions six and higher. Existing low-dimensional surveys tested only the flat metric in the tabulated basis, whereas the paper treats hh as an unknown and shows that allowing a general hh produces additional Lorentzian solutions, such as those on aff(1)⊕R\mathfrak{aff}(1)\oplus\mathbb{R}. A systematic classification and solution of the finite linear systems for general hh therefore remains incomplete.

References

Solvable algebras of dimension six and higher are not classified, and Appendix~\ref{app:F} (Table~\ref{tab:landscape}) collects the known solutions with $d\le11$. Even in dimension $\le5$, where the algebras are classified, the survey of Ref. tests only the flat metric in the tabulated basis; solving Eq.~eq:master for general $h$ on each classified algebra is a finite linear-algebra task that has not been carried out, and the $\mathfrak{aff}(1)\oplus\mathbb{R}$ example of Sec.~\ref{sec:radical} shows that it yields new Lorentzian solutions.

eq:master:

hab [ta,[tb,tc]]=0∀ c,h=CTη C.h_{ab}\,[t^a,[t^b,t^c]]=0\quad\forall\,c, \qquad h=C^{T}\eta\,C .

— Lie Algebra Saddles in the IKKT Matrix Model and Criteria for the Emergence of Time  (2609.40183 - Liao, 30 Sep 2026) in Section 6, Discussion, item (ii); Appendix F, Table 1